Università degli Studi di Palermo (UNIPA),
Dipartimento di Energia, Ingegneria dell'Informazione e Modelli Matematici (DEIM) Université du Québec
Institut National de la Recherche Scientifique (INRS) Centre Énergie, Matériaux et Télécommunications (EMT)
ULTRA-BROADBAND COHERENT DETECTION
OF TERAHERTZ PULSES VIA
CMOS-COMPATIBLE SOLID-STATE DEVICES
byAlessandro Tomasino
Thesis or dissertation submitted to obtain the double Degree of Philosophy Doctor (Ph.D.)
in Energy and Materials Science (EMT), Varennes, Quebec and
in Energia e Tecnologie dell’Informazione (DEIM), Palermo, Italy
Jury d’évaluation
Président du jury Prof. Alessandro Busacca
University of Palermo, Italy
Examinateur interne Prof. Roberto Morandotti
INRS-EMT, Québec, Canada
Examinateur externe Prof. Gianluca Giustolisi
University of Catania, Italy
Directeur de recherche Prof. Roberto Morandotti
INRS-EMT, Québec, Canada
Codirecteur de recherche Prof. Alessandro Busacca University of Palermo, Italy
This thesis is dedicated to my parents, my sister, and all my family,
Acknowledgments
I am really fortunate and grateful for the chance to have known my advisors, Prof. Alessandro Busacca at the University of Palermo and Prof. Roberto Morandotti at INRS-EMT, who have been foremost my mentors and inspiring figures. They taught me how to be not only a good scientist but also a well-rounded and eager to learn person, who firmly keeps in mind his targets. Thanks to them, I had the great opportunity to work on some very important and timely projects with the freedom to pursue some of my own ideas. Their unique insight into the concepts and their professional guidance have been continuously inspiring. The passion for science they demonstrated has addicted me and is responsible for my initial interest in the terahertz area. A special acknowledgment goes to Dr. Salvatore Stivala, who taught me numerous useful and practical skills, while I was taking the first steps within the experimental Optics, at the beginning of my Ph.D. studies. I also want to thank him for his valuable support and precious help in many others academic concerns.
I would like to thank Prof. Marco Peccianti for all those times we had very fruitful discussions, for his incredible open minded and creative approach, which has strongly amplified my excitement in chasing my objectives. His advice and help have been definitely invaluable. I wish to thank my colleagues, at both INRS-EMT and UNIPA, for their friendship and the great time we spent together in the last five years. Many pleasurable and thought-provoking discussions with them have served as a nucleation site for many new ideas and personal growth. I will treasure these memories and all the unforgettable experiences, for a lifetime.
I am grateful to those people who have been particularly close to me during the whole doctorate path and who have always encouraged and pushed me to give my best.
I also desire to dedicate a cordial thank to those who have left me an indelible memory, even though they either had to move away because of diverse reasons or simply wanted to drift apart eventually.
Last but not least, I wish to thank my parents and all my family, without whom my whole path would have been purposeless and meaningless.
Abstract
In this dissertation, we develop and demonstrate an innovative and fully integrated technique aimed at the coherent detection of terahertz (THz) radiation, i.e., electromagnetic waves whose frequency content conventionally falls in the spectral window ranging between 0.1-10x1012 Hz. We named such a detection technique solid-state biased coherent detection (SSBCD), since it is based on a solid-state medium and allows simultaneously recording both the amplitude and phase information of ultrashort THz pulses, i.e. featuring ultra-broadband spectra (> 10 THz). As such, our technique can be successfully used in those systems where THz pulses are employed as either diagnostic tool or signal carriers, such as in the areas of time-resolved spectroscopy and imaging or signal processing. SSBCD is based on platform fully compatible with the CMOS process, i.e. a microfabrication technology commonly employed for the realization of miniaturized electronics circuits (chips), thus being cost-effective and particularly reliable for the production of a great number of devices. Hence, its affordability makes it attractive for both a broad scientific and industrial audience. Indeed, the fundamental advantage of the technique presented here is the unlimited operating bandwidth in the whole THz range (allowed via interaction with ultrashort pulse durations), thus potentially addressing many of the issues and constraints of those THz solutions where the detection scheme represents a bottleneck in terms of the entire system frequency response.
Following a brief introduction regarding state-of-the-art of the THz technology and its different spectral regimes of operation, we will mainly review those detection techniques, which have been lately demonstrated to achieve the exact reconstruction of ultrashort THz transients. In particular, we will focus on those methods, which allow the detection of THz radiation, the spectrum of which covers the entire THz domain or even beyond (namely, the ultra-broadband regime). We will see that such particular techniques are essentially gas-based and rely on a similar concept, since the so far available solid-state methods, representing the state-of-the-art
in the THz detection area, are not suitable in the ultra-broadband regime, since they suffer a limited frequency response. Then, we will pass to the detailed description of mainly three different approaches, highlighting both advantages and drawbacks or limitations, eventually focusing our attention on the air-biased coherent detection (ABCD) technique. Indeed, we will show that our novel approach essentially overcomes some of the crucial issues of the ABCD method, by adopting particular, yet very common solid-state media (glasses) and plain integrated structures. ABCD exploits the nonlinearity of air and therefore operates at optical probe energies in the order of microjoules and bias voltages as high as several kilovolts. This restricts its application to expensive, bulky amplified ultrafast laser systems, and slow, high voltage amplifiers, which limit the noise performance and imply health hazard (electrical shocks). On the contrary, we show how the employment of CMOS-compatible dielectrics as nonlinear media, allows to dramatically decrease not only the requirement of optical energy to the level of nanojoules but also to greatly shrink down the size of the interaction region between the THz and optical pulses, with respect to the case of air. This results in the possibility to perform the THz detection in a compact structure, by using orders of magnitudes lower bias voltages, comparable to those regularly employed for photoconductive antennas. Such results pave the way to the realization of a unique and portable device that can be potentially driven by laser oscillators (featuring very good beam stability) and low-voltage amplifier, operating at much higher repetition rates and modulation frequencies, which will result in the significant increase of both dynamic range and signal-to-noise ratios.
Sommario
Il presente lavoro di tesi presenta e dimostra una tecnica innovativa e completamente integrata dedita alla rilevazione coerente di radiazioni a frequenze Terahertz (THz), cioè di onde elettromagnetiche il cui contenuto frequenziale cade convenzionalmente nella finestra spettrale compresa tra 0.1-10x1012 Hz. Tale tecnica è stata battezzata col nome di solid-state-biased
coherent detection (SSBCD), dal momento che essa sfrutta le proprietà di un mezzo a stato
solido e consente di ricostruire simultaneamente l’informazione sulla fase e sull’ampiezza degli impulsi THz, anche nel caso in cui quest’ultimi siano dotati di spettri a banda cosiddetta “ultra larga” (> 10 THz). Tale metodo di rivelazione può essere utilizzato con successo in quei sistemi in cui gli impulsi THz vengono comunemente impiegati come strumento diagnostico o come portanti di segnali a banda stretta, per esempio nelle aree della spettroscopia nel dominio temporale e nell’elaborazione di immagini o segnali, rispettivamente. La tecnica SSBCD si basa su una piattaforma completamente compatibile con il ben noto processo CMOS, cioè una tecnologia di micro fabbricazione comunemente impiegata per la realizzazione di circuiti elettronici miniaturizzati (chips), essendo quindi economicamente conveniente e particolarmente affidabile per la produzione di un gran numero di dispositivi per singolo processo produttivo. Queste caratteristiche rendono il metodo SSBCD particolarmente attraente per un ampio pubblico sia strettamente accademico sia industriale. Infatti, il suo principale vantaggio è rappresentato da una risposta spettrale estremamente larga, così da coprire l’intera gamma del dominio THz (qualora la durata temporale dell’impulso ottico lo consenta), permettendo potenzialmente di risolvere gran parte dei problemi e dei limiti che caratterizzano le attuali tecniche di rivelazioni, le quali rappresentano invece il collo di bottiglia di molti sistemi che operano a frequenze THz.
Di seguito, dopo una breve panoramica sulla tecnologia THz ed i diversi regimi spettrali di funzionamento, esamineremo le principali tecniche di rilevazione, che sono state recentemente
dimostrate per ricostruire nel modo più fedele possibile impulsi THz ultra corti. In particolare, ci concentreremo su quei metodi che consentono di ricostruire la forma di impulsi i cui spettri coprono l'intero dominio THz e anche oltre (vale a dire, il regime ultra-broadband). Vedremo che tutte queste tecniche sono essenzialmente basate sull’impiego di gas, sfruttando sostanzialmente lo stesso fenomeno fisico, dal momento che le configurazioni basate sui materiali a stato solido e che rappresentano lo stato dell'arte nell'area della rivelazione THz, non sono adatte per operare nel regime ultra-broadband, essendo caratterizzate da una limitata risposta in frequenza. Pertanto, passeremo alla descrizione in dettaglio di tre approcci diversi, evidenziandone vantaggi e inconvenienti, concentrando infine l’attenzione su un metodo di rivelazione detto air-biased coherent detection (ABCD). In effetti, mostreremo che il nostro nuovo approccio supera essenzialmente alcune problematiche cruciali della tecnica ABCD, grazie all’utilizzo di materiali alquanto usuali (essenzialmente dei vetri) e un’unica semplice struttura integrata. Infatti, il meccanismo intrinseco nell’ABCD sfrutta la non linearità dell’aria, richiedendo perciò energie ottiche dell'ordine dei microjoule e tensioni di polarizzazione pari a diversi kilovolt. Ciò restringe la sua applicazione da un lato a sistemi laser amplificati, che sono costosi e voluminosi, e dall’altro ad amplificatori ad alta tensione, che funzionano a basse frequenze di modulazione (onde quadre), limitando le prestazioni in termini di rumore, nonché comportando rischi per la sicurezza dell’operatore. Al contrario, mostreremo come il metodo SSBCD consente di ridurre drasticamente non solo il fabbisogno di potenza ottica al livello dei nanojoule, ma anche le dimensioni fisiche della regione di interazione fra impulsi THz ed ottici, rispetto al caso dell’ABCD, permettendo così l’utilizzo di livelli di tensione di polarizzazione paragonabili a quelli utilizzati regolarmente per le antenne fotoconduttrici. Tali risultati spianano la strada alla realizzazione di un dispositivo unico e portatile che può essere potenzialmente pilotato da oscillatori laser (che generano fasci laser di migliore stabilità ma a ben più basse energie) e amplificatori a bassa tensione, operanti a frequenze di modulazione molto più elevate, portando così ad un significativo aumento della dinamica dei segnali rivelati e dei loro rapporti segnale-rumore.
Résumé
Dans cette thèse, nous développons et démontrons une nouvelle technique entièrement intégrée ayant pour but de détecter la cohérence des rayons Térahertz (THz), c'est-à-dire des ondes électromagnétiques dont le contenu spectral est compris dans la fenêtre spectrale comprise entre 0.1-10x1012 Hz. Nous avons appelé cette technique de détection solid-state-biased coherent detection (SSBCD) puisqu'elle est basée sur un état solide et permet d'enregistrer simultanément des informations d'amplitude et de phase d'impulsions THz, même dans le cas de spectres avec ultra-large bande (> 10 THz). En tant que telle, elle peut être potentiellement utilisée dans les systèmes où les impulsions THz servent d’outil de diagnostic ou de supports de signaux, comme dans les domaines de la spectroscopie à résolution temporelle et de l'imagerie, ou encore du traitement du signal. La technique SSBCD est basée sur une plate-forme entièrement compatible avec un processus CMOS. CMOS est une technologie de micro-fabrication couramment utilisée pour la réalisation de circuits électroniques miniaturisés (chips), donc rentables et particulièrement fiables pour la production d'un grand nombre de dispositifs. Par conséquent, son accessibilité rend cette technologie attrayante à la fois pour le public scientifique et industriel. L’avantage principal de la technique présentée ici est la bande passante illimitée dans toute la zone THz (pour une durée d'impulsion laser fixée), permettant ainsi de résoudre tous les problèmes et contraintes de ces solutions THz où l'étape de détection limite principalement les performances du système entier. Après une brève introduction sur la technologie THz et ses différents régimes spectraux, nous passerons en revue toutes les techniques de détection qui ont été récemment démontrées pour obtenir la reconstruction exacte d’états transitoires THz ultra-courts. En particulier, nous nous concentrerons sur les méthodes qui permettent la détection de rayons THz dont les spectres couvrent tout le domaine THz, ou même au-delà (à savoir, le régime de bande ultra-large, c'est-à-dire plus de deux décades). Nous verrons que toutes ces techniques sont essentiellement basées sur les gaz et reposent sur un concept similaire, puisque
les configurations à l'état solide, représentant jusqu'ici la fine pointe du domaine de la détection THz, ne sont pas appropriées dans le régime de bande ultra-large, car elles souffrent d'une réponse en fréquence limitée. Ensuite, nous passerons à la description détaillée de trois approches différentes principalement, en soulignant les avantages ou les inconvénients et les limitations, et en concentrant finalement l'attention sur la technique appelée air-biased coherent detection (ABCD). En effet, nous montrerons que notre approche novatrice résout fondamentalement certaines des questions cruciales de la méthode ABCD, en adoptant des matériaux communs à l'état solide (verres) et des structures intégrées particulières. ABCD exploite la non-linéarité de l'air et fonctionne donc à l'énergie de la sonde optique de l'ordre du microjoule et de la tension de polarisation jusqu'à plusieurs kilovolts. Ceci limite son application à des systèmes laser ultra-rapides amplifiés coûteux et volumineux et à des amplificateurs à haute tension lents, qui limitent les performances de bruit et pourraient impliquer un danger pour la santé. Au contraire, nous montrons comment l'utilisation d'un matériau à l'état solide en tant que moyen non-linéaire permet de réduire considérablement non seulement l'exigence d'énergie optique au niveau des nanojoules, mais aussi la taille de la région d'interaction, par rapport à la technique basée sur l'air. Ceci engendre la possibilité d'effectuer la détection THz dans une structure confinée en utilisant des ordres de grandeur de tensions de polarisation inférieure, comparables à celles couramment utilisées pour les antennes photoconductrices. De tels résultats ouvrent la voie à la réalisation d'un dispositif unique et portable pouvant être piloté par des oscillateurs laser (très bonne stabilité du faisceau) et des amplificateurs basse tension, fonctionnant à des fréquences de modulation et de répétition beaucoup plus élevées, qui se traduira par l’augmentation significative de la gamme dynamique et des rapports signal sur bruit.
Content
Acknowledgments ... 5 Abstract ... 7 Sommario ... 9 Résumé ... 11 Chapter 1 ... 2 Introduction to THz radiation ... 2 1.1 Background ... 2 1.2 THz spectral regimes ... 4 1.3 Motivation ... 7 Chapter 2 ... 10Ultra-broadband THz coherent detection techniques ... 10
2.1 Dynamic Range and Signal-to-Noise Ratio ... 11
2.2 Nonlinear interaction of THz radiation and optical beams in gas media ... 15
2.2.1 Terahertz-Field-Induced Second Harmonic Generation ... 15
2.2.2 Phase-matching and Gouy phase shift ... 18
2.3 Air-breakdown coherent detection ... 21
2.4 Optically-biased coherent detection... 25
2.4.1 Analytical model ... 25
2.4.2 Experimental results ... 27
2.5.1 General aspects ... 30
2.5.2 Phase matching constraint ... 33
2.5.3 Analysis of the Gouy phase shift in the ABCD method ... 35
2.5.4 Dependence on the gas parameters ... 37
2.5.5 Plasma absorption ... 40
2.5.6 Dependence on the beam intensity profile ... 42
2.5.7 Balanced air-biased coherent detection ... 43
2.6 Noise evaluation in the ABCD technique ... 48
Chapter 3 ... 54
Silica-based Solid-State-Biased Coherent Detection of ultra-broadband THz pulses ... 54
3.1 Limitation of the ABCD technique ... 55
3.2 Previous demonstration of TFISH in solid materials ... 56
3.3 A CMOS process-compatible material: Silica ... 60
3.3.1 Simulations of the static electric field ... 63
3.3.2 SSBCD device microfabrication ... 66
3.4 Ultra-broadband THz time-domain spectroscopy set-up ... 67
3.5 Characterization of the first generation of SSBCD devices... 71
3.5.1 Bias voltage dependence ... 72
3.5.2 Probe energy dependence ... 75
3.6 Characterization of the second generation of SSBCD devices ... 77
3.6.1 Four-wave mixing in silica with THz radiation ... 79
3.6.2 Dependence on the slit width and electrode size ... 82
3.6.3 Spectral response ... 86
3.7 Comparison between SSBCD and EOS ... 87
Equation Chapter (Next) Section 1 ... 96
Chapter 4 ... 96
Beyond silica: Silicon nitride-based solid-state-biased coherent detection ... 96
4.1 SiN properties: differential THz-time domain spectroscopy ... 97
4.2 Four-wave mixing in silicon nitride ... 101
4.2.1 Fabrication of the SiN-based SSBCD device ... 103
4.3 Comparison with the ABCD technique ... 104
4.4 Characterization of the SiN-based SSBCD devices ... 104
4.4.1 Simulation of the field enhancement ... 104
4.4.2 Bias voltage dependence ... 110
4.4.3 Direct estimation of the THz electric field peak in the slit ... 113
Conclusions ... 116
Conclusions ... 122
Appendix A ... 128
Two-color plasma THz sources ... 128
Appendix B ... 136
A.1 List of articles published in scientific journals ... 136
A.2 List of conference proceedings ... 137
References ... 142
List of Figures ... 154
Chapter 1
Introduction to THz radiation
1.1 Background
Beyond its physical meaning as a frequency unit of measurement, with the term “terahertz radiation” we indicate electromagnetic waves the spectral content of which belongs to the intermediate portion between the highest region of the microwaves (millimeter and sub-millimeter waves) and the lowest part of the optical domain (far-infrared, FIR) [1], [2]. Alternatively, they are sometimes called T-ray, in analogy with X-rays, and conventionally their spectra span the two-decade wide range 0.1-10x1012 Hz (see Fig. 1.1), which corresponds to the wavelength range of 30-3000µm in vacuum (wavenumber spanning the range 3.3-333.3 cm-1). However, it is worth mentioning that some authors still talk about terahertz (THz) radiation, even though they deal with waves featuring frequencies partially covering the FIR (10-30 THz), just because such waves are being generated and detected via extremely wideband (several octaves) THz sources and detectors [1].
Although THz radiation constantly permeates our daily surrounding environment, due to the fact that it is incoherently emitted as part of the black-body radiation spectrum from any object with a temperature greater than few tens of Kelvin [3]–[5], we cannot directly experience its presence, because the interaction with matter is generally very poor. Indeed, THz radiation easily penetrates several optically opaque dry materials, whereas its photon energy (~ 4.1 meV) is too low for the typical energy transition mechanism (semiconductor band-gap based detection) exploited in optical technologies, which makes its handling quite cumbersome. For these reasons, this part of the electromagnetic spectrum was the least explored in the past, even though the very first experiment involving THz transients generated in quartz crystals [6] can be dated to the same period in which the second harmonic generation demonstrated in lithium
Figure 1.1 Subdivision of the electromagnetic spectrum. The THz domain is situated between microwaves and mid-infrared radiation (Optics).
niobate crystals, i.e. right after the laser was demonstrated (the 1960s). After that, the THz domain was mostly set aside for several decades and essentially no significant works appeared in scientific journals. This was mainly due to the poor interest and the bare knowledge of the actual mechanism behind such a type of radiation, with respect to the emerging optical domain, boosted instead by the development of the laser technology. Therefore, the previous lack of any reliable device operating in the THz range led to the expression “THz-gap”, to indicate the inability to realize standalone systems able to either efficiently generate or detect and in general manipulate information carried by THz radiation. Nevertheless, in the last two decades (roughly from the 1990s), tremendous effort has been made in the development of the THz technology [7] and its growth was so impressive that it rapidly resulted in a noticeable decrease and almost closure of such a gap. As a matter of fact, nowadays we better refer to it as “THz frontier”. The rapid advances in THz research can be attributed to the ongoing and parallel development of the field of optoelectronics and in particular of ultrafast chirped pulsed laser systems, able to emit optical pulses with femtosecond duration (down to the few cycles regime, Nobel prize in Physics 2018) [8]–[10]. A plethora of new and exciting opportunities in the area of the THz science and technology have soon emerged in many research areas, spanning from imaging and spectroscopy to microelectronics and telecommunications [11]–[14]. In addition, since THz radiation is recognized to be non-ionizing, THz technology has been arousing an ever-increasing interest in important fields such as (bio-) chemical sensing, monitoring, quality control, characterization of semiconductors and high-temperature superconductors, as well as medicine, forensic science, defence and security [11], [12]. In each of these research areas, THz
Figure 1.2. Chart of the THz generation techniques deployed as a function of the THz spectral emission and available average power. Compact terahertz sources exhibit low power and conversion efficiencies of much less than 1% in nearly every case, as the frequency rises into the terahertz range, the source output power plummets. Here, the Pf2 = constant line (being P the power and f the frequency), is the sort of power-frequency slope one would expect to see in a more mature RF device, while the Pλ = constant line (being λ the wavelength), is the expected slope for some commercial lasers. Higher frequencies are accessible by means of laser techniques only. (Figure adapted from Ref. [15]).
sources and detectors have been employed according to the specific application, in order to better accomplish the requested tasks. Historically, this has led to a natural differentiation of the THz products, mainly divided into pulsed and continuous wave (CW) solutions. This is explained in more details in the following sections.
1.2 THz spectral regimes
We typically distinguish among three THz spectral regimes of operation [1], depending on the main approaches that have been adopted so far in order to obtain efficient THz generation and detection, i.e. the frequency multiplication of microwaves occurring in nonlinear devices and
nonlinear optical methods involving ultrashort laser pulses [13], [14], as generally shown in Fig. 1.2. The type of classification that we are going to present is pretty useful in order to better comprehend the goals of this doctoral project. While THz radiation obtained by means of microwave techniques (up-conversion) results in a nearly single (sinusoidal) frequency (suitable, for instance, as a a carrier in transceiver systems), optical techniques
(down-conversion) usually allow for the generation and manipulation of coherent broadband THz
pulses. Therefore, all the techniques find a a place in the following three macro-categories:
Narrowband/Continuous Wave. High harmonic waves in diodes with strong nonlinear
current-voltage characteristics have also been demonstrated to be suitable for single-frequency radiation manipulation in the THz region (Fig. 1.3(a)). Quantum cascade lasers (QCLs), realized by the mutual coupling of multiple quantum wells are a common example for narrowband CW THz generation [16] (Fig. 1.3(b)). The quantum cascade arises from electrons tunneling between adjacent quantum wells. These electronic (intraband) transitions lead to the emission of photons with energies falling into the THz range. To date, QCLs have been the only feasible realization of the so-called “THz laser”, analogous to the optical counterpart. For sake of completeness, we mention that CW THz radiation can also be obtained via the nonlinear beating of two laser beams with slightly different central frequencies in suitable second-order material. The resulting beating envelope oscillates at a frequency equal to the difference between the two carrier beams, which falls in the THz range (Fig. 1.3(c)).
Broadband (0.1-4 THz). In contrast to the case of CW radiation, THz pulses featured by a
wideband spectrum (i.e. whose central frequency is of the same order of magnitude of its linewidth) can be generated (detected) through optical down (up) conversion of ultrashort laser pulses, whose temporal duration is sub-picosecond. Such a type of radiation is characterized by pulse durations of few picoseconds and typical quasi-single cycle temporal shape. They can be regarded as sinusoids modulated (i.e., apodized) by a short Gaussian envelope, which limits the number of cycles of the sinusoidal carrier. The larger the bandwidth, the shorter the Gaussian envelope and the lower the number of cycles of the THz carrier within the pulse duration. In this case, the most widespread generation techniques are optical rectification in quadratic media and photoconductive switches, whereas for detection, electro-optic sampling (EOS) [17]–[21]
Figure 1.3 Subdivisions of the THz generation methods. Continuous wave radiation featured by a multicycle shape can be achieved via either (a) microwave harmonic generation in nonlinear devices (mixer) or (b) integrated devices based on semiconductor nanostructured lattices (QCL). (c) Down-conversion from the optical domain in second-order materials, allow the generation or both pulsed and CW THz radiation. (Figure adapted from Ref. [1]
and photoconductive sampling (PCS) [22]–[29] represent the state-of-the-art techniques. (Fig. 1.2(c)).
Ultra-broadband (bandwidth larger than 10 THz). Beyond the regime of broadband radiation,
there is a relatively recent emerging area, where all the efforts are devoted to the handling of so-called single-cycle THz pulses. Essentially, such type of THz pulses are featured by exactly one carrier cycle, because of the extremely narrow temporal extension of the relative Gaussian envelope, hence the name ultrashort THz pulses. Their pulse duration is in the range of few hundreds of femtosecond or less and their spectral width can extend into the multi-THz regime. Indeed, in this case, the carrier frequency is lower in value than its linewidth (total bandwidth). For sake of completeness, it should be mentioned that similarly to the case of ultrashort lasers, the envelope of a single cycle THz pulse could be defined up to a phase term named pulse chirp
[8]. Ultrashort THz pulses are difficult to manage, because, as soon as they travel through the free space, the several absorption lines of the water molecules distort the pulse shape, which then shows long-lasting residual oscillations. Moreover, most of the common techniques prevent achieveing a pure single-cycle THz pulse, because of some inherent and detrimental interactions with the nonlinear materials involved in either the generation or detection scheme, as will be better explained in the following sections. Particularly, it is worth highlighting that, currently, there is no metrological system, which allows to directly access the actual form of the THz pulses as emitted by a particular source. Indeed, any detection method features a precise frequency response, which is not simply flat all over the bandwidth, thus affecting the reconstructed THz spectrum and the associated THz transient shape. For this reason, we would like to clarify that with the term ultra-broadband THz pulses, which all along underlines this thesis work, we strictly indicate those techniques which reduce as much as possible any spectral artefact, thus allowing the retrieving of THz spectra which are virtually modulation-free and continuous, i.e. do not show notch frequencies or stop bands. Nowadays, plasma-based techniques are mainly exploited in the ultra-broadband regime for both generation and detection [30]–[42].
1.3 Motivation
To date, effective methods have been developed for the cases (i) and (ii), which allow for the manipulation of THz sources [7-12]. Various promising solutions for the generation of high power broadband THz pulses have been proposed, differing in their complexity, THz bandwidth, efficiency, and footprint. However, the realization of a practical ultra-broadband detector in the THz domain is still a key challenge. Commonly, direct detectors based on thermal absorption have been employed, such as helium-cooled silicon and germanium bolometers [1]. Most of these solutions require elaborate cooling systems to reduce thermal background. However, more importantly, they are incoherent detectors, i.e. not sensitive to the waveform of the THz radiation. This limits their use in field-resolved experiments, such as time-domain spectroscopy. EOS and PCS are both able to simultaneously recover the amplitude and phase, but their deployment is limited to the broadband regime only. This is because they require the use of solid-state materials, particularly compound semiconductors, which unavoidably suffer
detrimental phenomena such as non-negligible free-carrier lifetimes, phonon resonances, phase-mismatch and etalon effects within the THz domain. These concomitant effects can lead to significant dips in the spectrum, thus resulting in the detection of suitable THz bandwidths of only a few THz (mainly 0.1-4 THz in state-of-the-art systems) [19], [25], [43]–[48]. The need for a robust detection paradigm able to address all the above-mentioned issues and which could represent an ultimate solution for many THz systems, currently seems to be highly pressing and demanding. Indeed, the ultra-broadband regime is particularly attractive due to the advantages which such a wide spectral range has with respect to conventional THz systems [49]. On one side, a 10 THz-wide radiation lasts only a few hundreds of femtosecond (full width at half maximum), enabling high-resolution time-of-flight measurement for, e.g., 3D THz imaging of multilayered structures [50] or thickness evaluation of thin films [51], [52]. On the other side, many substances such as semiconductors [53], liquid crystals [54], chemicals, like drugs and explosives [55], and biopolymers like proteins and DNA [56], [57], possess specific roto/vibrational modes above 2 THz. Therefore, the possibility of providing ultra-broadband detection capabilities is essential for their complete investigation in a wider THz spectral window [58]. Most of the detection methods able to properly operate in the ultra-broadband regime are mainly gas-based techniques, which imply the use of experimental scheme and equipment relatively more complex and expensive with respect to those employed in either PCS or EOS techniques. In this work, we present all the efforts made in the development of a compact, standalone, solid-state device, which can be easily realized and employed as general purpose THz detector, in both broadband and ultra-broadband regimes.
Chapter 2
Ultra-broadband THz coherent detection
techniques
In the second chapter of this dissertation, we will review the most important techniques proposed until now for the ultra-broadband coherent detection of THz pulses. As we have already mentioned in the first Chapter, although EOS and PCS are considered the state-of-the-art in the THz detection area, they do not allow for ultra-broadband operation according to the definition previously given. Therefore, here we will not detail the working principles of such detection methods, referring the interested reader to the specific literature [1], [17], [18], [24]. Instead, we will mainly focus the attention onto gas-based techniques. Gases show resonant-free frequency response even in the THz domain, due to the lack of any periodic arrangement of the constituting particles. Moreover, a gas continuously regenerates itself due to the random movements of the molecules, thus allowing for nonlinear interaction among high-intensity laser beams, without the risk of permanent damage. Air is usually the most common employed gas, because of immediate availability and very low chromatic dispersion, which is a desirable property while handling ultra-broadband radiation. In addition, since air is composed of 78% of nitrogen, it already exhibits remarkable performance in the detection of THz pulses. However, higher efficiencies can be achieved by using other types of gases such as pure nitrogen, argon, xenon, and krypton, as will be discussed in the following. All the gas-based schemes share the same background mechanism, which somehow resembles the so-called electric-field-induced second harmonic (EFISH) generation [59]–[62], occurring in centrosymmetric media, which will be described in more details in a dedicated section. However, since the nonlinear interaction between an optical beam and a traveling THz electric field wave results in crucial differences
with respect to the case of a purely static electric field, such a phenomenon is more correctly referred to as Terahertz-Field Induced Second Harmonic Generation (TFISH). Before starting with an overview regarding the current THz detection techniques, it is essential to discuss the concepts of dynamic range and signal-to-noise ratio, which will be recalled several times throughout the text. These two parameters are the most widespread noise figures employed for an exhaustive description of the quality of a THz-TDS system, yet there is a quite great confusion in literature regarding the real meaning and the difference between these two quantities.
2.1 Dynamic Range and Signal-to-Noise Ratio
Since the efficiency of the typical THz generation and detection methods is still far from the theoretical quantum limit reachable in the mechanisms involving optical beams (~10-2) [63], the noise affecting the detection of THz radiation plays a crucial role in the ultimate determination of the performance of THz-TDS solutions. Among others, the main parameter often provided with THz systems is the dynamic range (DR). Another important parameter, somehow similar to the DR and aimed at describing the quality of a reconstructed THz pulse, is the signal-to-noise ratio (SNR). However, there are few considerations to point out before directly passing to the operating definition of both parameters. First, given the confusion that certain authors working in the THz community have raised until now, it is worth clarifying that DR and SNR are not the same parameter and do not exactly return the same type of information. Likewise, typical values for the two quantities are generally much different in terms of order of magnitude. Two more appropriate general definitions have been given by Naftaly et al. [64], as stated by the following sentences:
The DR describes the maximum quantifiable signal change. The SNR indicates the minimum detectable signal change.
The two definitions above can be interpreted in a more practical way. The DR expresses how much we can attenuate the signal and still be able to resolve it from the noise floor. The SNR, instead, expresses how much we can distinguish between the signal changes and the fluctuations induced by the noise. In light of these definitions, it is clear that the two parameters aim to
characterize the measured data in two different, yet complementary, ways. Another main point to highlight is that for a typical THz-TDS system (and, in general, also for other systems) both DR and SNR are not properly defined for either one single generation or detection scheme. Those values depend on the particular combination of source and detector, since the performance obtained for the same generation technique can be different if the associated detection approach is varied, and vice versa. Quantitatively, DR and SNR are evaluated as [64]:
max of the temporalamplitude
DR t
RMS of noise floor
(3.1)
mean value of temporal amplitude
SNR t
RMS of the temporal amplitude
(3.2)where RMS stands for root-mean-square. When analyzing a typical THz waveform, it is found out that both DR and SNR are not constant as a function of time, but they are usually higher for those time instants where the signal assumes its maximum value. Therefore, as suggested in [64], in order to evaluate optimum parameters, which can well reproduce the quality of the measurements, Eqs. (2.1) and (3.2) should be evaluated at the peak of the THz waveform. The strategy is the following:
DR: the delay between the two pulses is varied in such a way to record the noise floor before the arrival of the THz transient. Then, the peak (either positive or negative) of the THz waveform is measured and is then divided by the RMS of the noise floor.
SNR: the delay is fixed on the time position corresponding to the peak of the THz waveform. Several data points are acquired (in a number of ten or more) and by applying statistical methods to this dataset, the mean value of the peak and its standard deviation are evaluated. The ratio between such two quantities returns the SNR value.
As an example, these two procedures were used to analyze the dataset provided in Fig. 2.1(a). While one scan is in principle enough to evaluate the DR, it is necessary to iterate the measurement several times in order to access the SNR information. Therefore, the main THz
Figure 2.1 (a) Typical THz time domain waveform (blue line) reconstructed as the mean of nine consecutive temporal scans (left vertical axis). The standard deviation of such measurements is plotted as the purple line. The green curve represents the SNR plotted as a function of time (right vertical axis). (b) DR (blue cure, left vertical axis) and SNR (purple curve, right vertical axis) evaluated in the frequency domain. Note the different scale of the two vertical axes. (Figure adapted from Ref [64]). waveform shown in the figure is actually the mean of nine consecutive scans. The purple line shows the standard deviation (STD) of the set of data multiplied by 20, revealing that while STD assumes negligible values before the arrival of the THz transient, it greatly varies once the THz pulse shows up, assuming the highest value when the signal approaches its maximum amplitude (negative, in this case). The resulting DR is equal to ~10000 for this measurement. Moreover, the point-to-point ratio between the blue and purple curve returns the SNR as a function of time (green curve), with a maximum around 150. It is worth stressing that the two numbers are profoundly different (DR/SNR ~ 80), thus indicating that they represent different features of the measured THz waveform.
The definitions given above allow evaluating the DR and SNR in the time domain. Nevertheless, it is possible to retrieve the same parameters in the frequency domain as well. However, it is important to highlight that there is not a specific relationship between temporal and spectral DR and SNR, and most importantly, they assume different values for the same THz system. Since in a THz-TDS system, we can actually retrieve the temporal waveform only, whereas the spectra are subsequently calculated by numerical Fourier Transformation (FFT), the procedure for the evaluation of DR and SNR in the frequency domain is the following. First, one calculates the FFT of a dataset of measurements (in a number of tens or more). Then, statistical methods are
employed to evaluate its mean value (and the corresponding noise floor) as well as the standard deviation. As before, in principle, one measurement is sufficient for DR evaluation only. Finally, the following formulas are used:
mean FFT amplitude
DR f
noise floor
(3.3)
FFT mean value SNR f STD (3.4)By again considering the previous example, the application of Eqs. (2.3) and (3.4) leads to the curves plotted in Fig. 2.1(b). The DR as a function of the frequency closely reproduces the shape of the actual THz spectrum (note the natural scale on the left vertical axis). The maximum DR is around 2000. Conversely, the maximum SNR is around 70. As stated before, such numbers are not related to the homologous retrieved in the time domain. In addition, the ratio DR/SNR~30 reiterates the fact that the two parameters point out a different type of information. To this point, Fig. 2.1(b) helps in the proper interpretation of DR and SNR, as anticipated in the notes before. Indeed, on the one hand, the DR features a bell-shaped curve, centered on 0.5 THz, which quickly decays to values lower than 200 for frequencies higher than 1.5 THz. This means that the THz system under characterization is more sensitive to the frequencies nearby those corresponding to the maximum DR. In other words, if the THz signal is attenuated by around 2000 times, at least that frequency region can be still extracted from the noise and detected. On the other hand, the SNR shows a flatter frequency response and it is interesting to observe that its values are still close to the maximum (~60) at frequencies for which, instead, the DR is already one order of magnitude lower than its peak. Therefore, the SNR does not really depend on the DR and vice versa, which implies that if one attenuates the signal strength, its DR range changes, yet the SNR remains essentially unvaried. We also note that, once the DR approaches values two orders of magnitude smaller than its peak, the SNR is still around 25% of such a value. This means that this particular THz system is able to effectively resolve small variation of the signal over noise fluctuations, in the portion of the spectrum where the signal itself has a poor frequency content.
only to characterize the noise performance of our ultra-broadband set-up, when detection is performed via either ABCD or our new technique. This because the SNR is greatly affected by the stability of the THz source for the time instants where the THz pulse is present, due to the averaging among several contiguous measurements. On the contrary, the DR can be evaluated by knowing only one temporal waveform and the stability on the THz source does not play a crucial role, thus allowing to better highlight the properties of the detector stage.
2.2 Nonlinear interaction of THz radiation and optical beams in gas media
2.2.1 Terahertz-Field-Induced Second Harmonic Generation
Detection based on a centrosymmetric (χ(3)) material, as for the case of EOS, consists in the up- conversion of the THz frequencies to the optical domain, via the THz-driven frequency doubling of an optical probe beam. The so-generated new optical beam encodes the information carried by the THz wave and can be easily recorded via standard optical detectors, thus overcoming the difficulties affecting direct detection of THz radiation. As previously mentioned, TFISH essentially works as EFISH. In the latter case, a static electric field breaks the symmetry of the centrosymmetric medium, thus inducing an electrically-driven second-order behavior. Under this condition, an optical probe pulse propagating through such a symmetry-broken material, experiences a frequency doubling via the effective second-order coefficient driven by the static electric field (χeff(2) = χ(3) EDC). Similarly, in the TFISH case, since the THz electric field can be
regarded as static within a complete cycle of the optical wave (ωTHz << ωP), when the THz
beam is overlapped in both time and space with the probe, a third beam approximatively at the second harmonic (SH) frequency of the probe beam is generated [65]–[67]. This SH beam can be acquired, after proper probe filtering through a narrow-band filter (NBF), for example by means of a photodiode or a photomultiplier tube (PMT), as sketched in Fig. 2.2(a). In a practical case, a femtosecond NIR probe beam (λP = 800 nm) is approximatively converted via TFISH
into a λSH ~ 400 nm (blue) beam. In such a framework, the SH electric field ESHTHz and the
corresponding intensity ISHTHz of the THz-induced SH beam change proportionally to the THz electric field strength
E
THz and its intensityI
THz, respectively, according to the following two relations:Figure 2.2 (a) TFISH generation in air: the probe pulse is frequency-doubled when the THz beam is strongly focused around the focal spot of the optical beam. A photomultiplier tube acquires the filtered second harmonic beam. Simplified diagram representing the two possible energy transitions taking place in the four-wave mixing process giving rise to (b) difference frequency generation (DFG) and (c) sum frequency generation (SFG). (3)
THz SH P P THzE
E E E
(3.5)
(3)
2 2
THz SH P THzI
I
E
(3.6)where EP and IP are the probe electric field and intensity, respectively. For the sake of
completeness, it is worth mentioning that the process described by Eq. (2.5) and (3.6) is an approximation valid in the weak probe intensity regime only, since for higher intensities many others nonlinear phenomena could occur (such as multiphoton absorption, probe spectrum broadening, and others). Moreover, the Kerr coefficient itself could manifest a certain dependence on the optical energy. Such effects could give rise to a saturation of the actualISHTHz trend, which eventually results deviating from the predicted quadratic dependence on the THz electric field. Another great simplification in writing Eqs. (3.5) and (3.6) arises from assuming that we deal with a nonlinear interaction taking place among plane waves, in the absence of any phase-matching condition. Indeed, since the bandwidth of the THz pulse is not negligible, above all for ultrashort pulses, the generated TFISH beam is not exactly oscillating at the SH frequency. Rather, the overall interaction can be better explained in terms of a four-wave mixing (FWM) process driven by the third-order susceptibility (Kerr). Theoretically, as shown in Fig. 2.2(b)-(c), two possible FWM mechanisms could occur, i.e. either a difference frequency process (DFG) or a sum frequency generation (SFG). In the former case, two probe photons combine together to generate one THz photon and one photon at the DFG frequency
Figure 2.3. (a) Gouy phase shift experienced by the THz beam while crossing the focal point. The π-rotation results in a flip of the polarity of the THz pulse. zT and zR are the Rayleigh lengths of THz and
probe beam, respectively. Gouy phase shift for the case (b) DFG and (c) SFG generation process, along the propagation direction z with origin on the focal plane, evaluated for the case of zT = 0.8 mm and zR
= 4.8 mm. While for the DFG the total phase change (blue curve) never overcomes the value of ±π, thus promoting the coherent summation of the contributions to the total DFG beam in a wide range of propagation values, for the SFG case, the total phase change overcomes ±π-rotation for values relatively close to the focus, thus limiting the total conversion efficiency. (Figure adapted from Ref. [35]).
(DFG THz 2P), whereas in the latter case, two probe photons and one THz photon combine together to generate one SFG photon (
SFG
2
P
THz). It is worth noticing that both the DFG and SFG frequencies are actually quite close in value to the SH of the probe beam, becauseωTHz << ωP, but the phase-matching conditions for the two cases are profoundly different.
Indeed, it is well known that when a tightly focused Gaussian beam travels for several Rayleigh lengths (zR = kwP2/2, where k is the wave vector and w the beam waist on the focal spot) across
its focus, it experiences an additional phase rotation of π radiants with respect to a propagating plane wave featured by the same wavelength. Such an effect is known as Gouy phase shift [68] and essentially leads to a flip of the THz pulse polarity, as schematically depicted in Fig. 2.3(a). As will be better clarified in the next notes, this further phase contribution favors the DFG process against the SFG when air or other gases are employed as nonlinear materials, because of the long interaction length, necessary to achieve sufficient EFISH power conversion [37]. Therefore, from now on, the SFG will be totally neglected within the text. Moreover, since the THz bandwidth is generally much smaller than that of optical frequencies, it results ωSH ≈ ωDFG,
therefore we will often use equally the term DFG or SH to refer to the intensity of the TFISH beam.
2.2.2 Phase-matching and Gouy phase shift
In order to understand how the focusing of both probe and THz beams affects the overall efficiency of the DFG process, we assume that their electric field functions are well described by standard Gaussian profiles, which are expressed, in cylindrical coordinates, by the following relations:
2
2
( )
,
exp
,
1
/
1
/
P
P R P RE z
r
E r z
iz z
w
iz z
(3.7)
,
0,
1
/
THz THz THzE
E
r z
iz z
(3.8)where, r and z are the transversal and longitudinal coordinates, and the terms EP(z), ETHz0 and
zR, zTHz are the longitudinal amplitude profiles and the Rayleigh lengths of the probe and THz
beam, respectively. Therefore, the Gouy phase shifts of the two beams are given by:
arctan ,
arctan THz R THz R z z z z z z (3.9)Since common values for the THz spot size are in the order several hundreds of micrometers, with respect to the few tens of micrometers for the probe beam, the transversal dependence of the THz beam upon r can be neglected in Eq. (2.8) with a good approximation. Considering the slowly varying envelope approximation, the second harmonic beam featured by the electric field
ESH is generated according to the following nonlinear Helmholtz equation:
2 2 (2) 2 216
2
exp
,
SH SH T SH eff PE
ik
E
E
i kz
z
c
(3.10)where,
k
2
k
P
k
SH is the wave vector mismatch between the probe and SH beams, which is inherently negative ( k 0, as it can be easily understood by looking at the typical dispersion relations). Here, the THz-induced effective second-order susceptibility is expressed by:(2) (3) *
. .
where the asterisk sign stands for the complex conjugate of the THz electric field (DFG). The trial solution for ESH is determined for similarity to the fundamental field (Eq. (3.7)) as:
2
2
( )
2
,
exp
.
1
/
1
/
SH
SH R P RE
z
r
E
r z
iz z
w
iz z
(3.12)By inserting Eqs. (3.12), (3.7) and (3.8) in Eq. (3.10), it is found out:
4
(3) 2 *
,
,
SH P THz Pi
E
z
E E J
k z
n c
(3.13)where nP is the refractive index at the probe frequency, whereas the term
J
k z
,
has the form:
,
exp
'
', 1 '/ 1 '/
R T i kz J k z dz iz z iz z (3.14)and can be easily evaluated with a contour integration (Jordan’s Lemma, 2nd formulation) [69].
Finally, Eq. (3.13) becomes:
8 2 (3) 2
, exp , THz R THz SH P THz THz P R THz z z i E r z E E kz n c
z z (3.15)while the total Gouy phase shift which ESH undergoes is given by:
4
,
SHz
Rz
THzz
(3.16)where the double sign identifies the DFG (-) or SFG (+) generation process, respectively. Figures 2.3(b)-(c) show the phase trends expressed by Eq. (3.16) for the case of zT = 0.8 mm
and zR = 4.8 mm, which are typical focusing conditions of the THz and probe beams. We are
now able to comprehend while the Gouy phase shift favors DFG against SFG. Indeed, while the two propagating pulses nonlinearly interact to generate IDFG, the total phase change
4
DFG z R z THz z does not reach the value of ±π in a long range of propagation
(~cm). This means that each contribution to IDFG is accordingly summed in phase, leading to a
constructive interference, which enhances the DFG yield. Conversely, for the SFG case,
4
DFG z R z THz z
to the focal plane (~mm), beyond which the polarity of the THz pulse is reversed. This leads to a destructive interference, which effectively limits the total ISFG. By carefully examining Eq.
(3.15), we can notice that the total strength of the SH electric field depends not only on the THz strength and probe intensity but also on the interplay between the THz and probe Rayleigh lengths as well as the phase-matching condition between the probe and the SH frequencies. It is worth noticing that the Rayleigh length of the probe beam does not directly affect the phase-matching condition. In order to minimize the effect of phase-mismatch, the exponential term in Eq. (3.15) suggests that zTHz should be as short as possible. However, due to the long
wavelengths associated with the THz frequencies, this value has a relatively high lower limit (~1 mm), which eventually could result in poor convention efficiencies. On the other hand, since
zR is generally much larger than zT, the second ratio in Eq. (3.15) has an asymptotic limit equal
to zT itself, thus leading to an opposite trend with respect to the exponential term. Hence, a
trade-off clearly exists. Additional considerations about this point will be presented in the following sections, where other parameters playing a role in the overall detection will be taken into account. For example, as better explained later in the text, while the Gouy phase shift is actually limiting the performance of gas-based techniques, it does not play any particular role in our solid-state detection technique, yet based on the TFISH process. This because, in the latter case, the interaction is limited to a region extremely close to the focal plane (in the best case, only a few micrometers), before the THz pulse could undergo polarity inversion. As already stated in Eq.(3.6), it is clear how the PMT readout signal, which is sensitive to ISH only,scales with the
square of ETHz. This means that at this point, THz pulse detection is still incoherent, i.e. the
phase information is lost. Since most of the pulsed THz systems are based on the simultaneous detection of both amplitude and phase of the THz transient, i.e. coherent detection, several efforts have been devoted to the extraction of the phase information from the TFISH signal. Three main solutions have been proposed across the years, which somehow relies on a unique idea: the TFISH electric field beats with a so-called local oscillator ELO, i.e. a further electric
field oscillating at the same frequency of the probe SH, and then the cross term is isolated and detected. At first instance, such techniques essentially differ in the strategy adopted to generate the local oscillator and are named: air-breakdown coherent detection, optically-biased coherent detection, and air-biased coherent detection. It is worth mentioning that a specific feature
Figure 2.4 (a) Sketch of the experimental set-up employed in Ref [70]. The THz wave is generated by the nonlinear mixing of the fundamental and its second harmonic beams occurring on the first plasma channel. A couple of 90 off-axis mirror collimates and (after separation from the optical beams), focuses the THz beam in air together with the probe beam. The latter is focused via an optical lens and passes hrough a hole made in the middle of the mirror. The EFISH beam is hence formed and acquired by a PMT, after filtering the remaining probe power. (b) Time-resolved EFISH transients measured at three different probe intensities (increasing moving from the upper to the lower): 1.8x1014 W/cm2,4.6x1014 W/cm2,9.2x1014 W/cm2. Gradual conversion of the transient from unipolar to bipolar is observed. (Figure adapted from Ref [70]).
actually underlies the last technique, i.e. the heterodyne scheme, which provides a unique advantage against the other two methodologies, in terms of both noise figure and fidelity of the reconstructed coherence, as detailed in the following.
2.3 Air-breakdown coherent detection
Perhaps the very first attempt of implementing a gas-based ultra-broadband detection scheme, the air-breakdown coherent detection method was demonstrated both analytically and experimentally by Dai et al. [14-15]. In such a work, authors generated a train of ultra-wideband THz pulses via a two-color plasma technique (see Appendix A), realized by focusing a 800 m, 120 fs, 800 uJ, 1 kHz pump beam with its second harmonic, the latter achieved in a 100-μm- thick beta barium borate (β-BBO) type-I crystal, as sketched in Fig. 2.4(a). The THz wave is separated from the pump by interposing a silicon wafer -acting as a long pass filter- and then
Figure 2.5. (a) Intensity of the local oscillator ILOSH as a function of the probe intensity (black dots). Note that both the scales are logarithmic. The corresponding Keldysh parameter is shown on the top x-axis. It is possible to identify three main regions, separated by the vertical dashed lines, which indicate the probe intensity values for which the incoherent, hybrid and coherent detection regime are achieved, respectively. (b) Power spectrum of the THz waveform recorded in a nitrogen environment for a probe intensity of 9.2x1014 W/cm2 (shown in the inset). Some spectral artefacts appear in the spectrum, the actual origin of which is not totally understood. (Figure adapted from Ref [70]).
tightly refocused by a 90° off-axis parabolic mirror. A probe beam (obtained by splitting part of the main beamline and the polarization of which can be controlled by means of a half plate) is propagated through a hole made in the center of the last parabolic mirror. It is then focused through a lens the focus of which is placed on the same focal point of the parabolic mirror, thus being spatially overlapped with the THz spot. By scanning the time delay between the THz and the probe beam, it was possible to reconstruct the waveform of the SH intensity readout by means of the PMT. The results are shown in Fig. 2.4(b). While the intensity of the probe beam was below the value of 1.8x1014 W/cm2, the authors observed the detection of the TFISH beam
only, i.e. a signal shaped as the intensity of the THz waveform. However, as the probe intensity increased, they first noticed a change of the THz waveform from a unipolar to a hybrid bipolar curve for intermediate probe intensity in the order of 3.3x1014 W/cm2. This eventually turned into a biploar THz transient for intensities greater than 5.5x1014 W/cm2, which somewhat resembles the typical few-cycle waveform recorded through EOS in ZnTe crystals. Such experimental results can be explained with the aid of a simple mathematical model. We recall
that the readout signal of the PMT when only the THz pulse is present, is expressed by Eq. (2.1), which results in the upper transient in Fig. 2.4(b). However, when the probe beam becomes more intense, it starts to ionize air, giving rise to white light as results of self-phase modulation and self-steeping. The electric field associated to the frequency component at the second harmonic of the probe beam-induced plasma then acts as local oscillator (LO)
E
LOSH,which beats with the TFISH term. Such a mixing gives rise to a total DFG intensity, which consists of three main contributions:
2 (3)
2 2
(3)
2
22 cos
SH SH SH
DFG total P THz P THz LO LO
I E I E I E E E (3.17)
where θ is the phase difference generally present between the TFISH term and the LO component, which can be considered approximately constant since the only source for the LO field is the white light generated by the probe beam. We notice that the cross term in Eq. (3.17) is linearly dependent on the THz electric field, thus constituting the key factor for the detection of the full THz waveform. The third term is an offset contribution from the LO term and is easily filtered out by carrying out the detection via a lock-in amplifier synchronized with the chopping frequency of the THz beam. Therefore, Eq. (3.17) simplifies in:
(3)
2 2
(3)
2
2 SHcos
DFG P THz P THz LO
I I E I E E (3.18)
By means of Eq. (3.18), we are now able to understand the reason behind the change of the waveform nature presented in Fig. 2.4(b). Indeed, below the plasma threshold
I
Pth, the contribution of LO to the total SH intensity is negligible and Eq. (3.18) reduces to Eq. (3.6). However, once the intensity of the probe beam is high enough to generate plasma, the cross term becomes dominant with respect to the pure TFISH term, resulting in:
(3)
2
2
cos
th P P SH DFG I I P THz LOI
I
E
E
(3.19)Therefore, in the limit of a very high probe intensity, the readout signal of the PMT linearly scales with the THz electric field and the detection scheme becomes quasi-coherent. The latter condition explains the appearing of negative lobes in the intermediate curve in Fig. 2.4(b), which becomes more prominent in the lowest transient. The transition from the incoherent to the
coherent detection regime can be identified by taking into account the Keldysh parameter γ, which indicates the type of ionization process (either multiphoton or tunneling) occurring for a certain optical intensity. In particular, by measuring only the background offset intensity of the local oscillator SH
SH
2LO LO
I E as a function of the probe intensity (recorded by modulating the probe beam and blocking the THz beam), it is possible to build the graph shown in Fig. 2.5(a). Three main regions can be recognized, as highlighted by the vertical dashed lines, corresponding to the regimes of incoherent, hybrid and quasi-coherent detection, each of them characterized by different slopes. For low intensities, the probe beam generates no significant blue light (LO). As soon as the intensity overcomes the value of ~1.8x1014 W/cm2 (roughly the
plasma formation threshold), ILOSH starts to dramatically increase of several orders of magnitude, because of the multiphoton ionization, until approximately 5.5x1014 W/cm2. Beyond such a value, the further increase of the plasma carrier density is dominated by tunneling ionization (as pointed out by the condition γ < 0.5, holding for femtosecond laser pulses) and correspondingly the increase of ILOSH slows down. Under this condition, the plasma blue component is so strong that totally dominates over TFISH, and Eq. (3.19) can be satisfied. The coherence of this type of ultra-broadband detection method strongly depends on the amount of probe intensity available in the experiments. In particular, a fully coherent THz transient can be achieved at the expense of a significant portion of optical power split from the main beamline, necessary to induce the tunneling ionization regime. Moreover, such high probe intensities do not allow to hold the approximation of four-wave mixing (FWM) process in order to exhaustively describe the detection mechanism. Indeed, although the linear dependence on the THz electric field expressed by Eq. (3.19) maintains in a relatively large range of THz strength values (0-10 kV/cm), it has been demonstrated that the PMT signal recorded for increasing probe intensity significantly deviates from the theoretical quadratic trend. Rather, it shows a sort of saturation for the intensities able to trigger the tunneling ionization. Besides the nonlinear absorption induced by highly dense plasma channels, this effect can be ascribed to the fact that χ(3) could not be considered constant for such high values, thus degrading the efficiency of the FWM process. Additionally, the requirement of elevated ILOSH in order to ensure the coherent detection, seriously limits the noise performance of the detection method because of the tremendous
